A Gauss–Jacobi Kernel Compression Scheme for Fractional Differential Equations

Abstract

A scheme for approximating the kernel w of the fractional $\alpha$-integral by a linear combination of exponentials is proposed and studied. The scheme is based on the application of a composite Gauss–Jacobi quadrature rule to an integral representation of w. This results in an approximation of w in an interval $[\delta ,T]$, with $0<\delta$, which converges rapidly in the number J of quadrature nodes associated with each interval of the composite rule. Using error analysis for Gauss–Jacobi quadratures for analytic functions, an estimate of the relative pointwise error is obtained. The estimate shows that the number of terms required for the approximation to satisfy a prescribed error tolerance is bounded for all $\alpha \in (0,1)$, and that J is bounded for $\alpha \in (0,1)$, $T>0$, and $\delta \in (0,T)$.

Appendix: A Technical Lemma

Estimate (4.7) of Lemma 4.2 relies on the infimum of

$$\begin{aligned} R_J\!\left( \ell \right) =\!\left( 3-\ell \right) ^{-1}\!\left( \ell +\sqrt{\ell ^2-1}\right) ^{-2J}\ , \end{aligned}$$

(A.1)

for $\ell \in (1,3)$. It is a simple exercise to show that for each $J\ge 1$, $R_J$ has a unique minimum point in (1, 3) and to estimate the asymptotic behavior of that minimum, at the limit $J\rightarrow \infty$. This is summarized in the following lemma.

Lemma A.1

For each $J\ge 1$, $R_J$ has a unique minimizer $\ell _J$ in (1, 3) given by

$$\begin{aligned} \ell _J=\frac{3-\mu \sqrt{8+\mu ^2}}{1-\mu ^2}, \quad \mu =\frac{1}{2J}. \end{aligned}$$

(A.2)

In particular $\ell _J\in (3/2,3)$, and at the limit $J\rightarrow \infty$, there holds

$$\begin{aligned} R_J\!\left( \ell _J\right) \sim \frac{\mathrm {e}}{\sqrt{2}}\, J\!\left( 3+\sqrt{8}\right) ^{-2J}. \end{aligned}$$

(A.3)

Proof

The derivative of $R_J$ is given by

$$\begin{aligned} R_J'\!\left( \ell \right) =\!\left( 3-\ell \right) ^{-2}\frac{\!\left( \ell +\sqrt{\ell ^2-1}\right) ^{-2J}}{\sqrt{\ell ^2-1}} \!\left( \sqrt{\ell ^2-1}-2J\!\left( 3-\ell \right) \right) . \end{aligned}$$

(A.4)

Thus, the critical points of $R_J$, satisfy

$$\begin{aligned} \mu \sqrt{\ell ^2-1}=3-\ell , \quad \mu =\frac{1}{2J}. \end{aligned}$$

(A.5)

It is therefore clear that $R_J$ has a unique critical point and that this point is the unique minimizer of $R_J$ in (1, 3). By squaring the last equality, we find that $\ell =\ell _J$ is a solution of the following equation

$$\begin{aligned} \!\left( 1-\mu ^2\right) \ell ^2-6\ell +9+\mu ^2=0. \end{aligned}$$

(A.6)

This equation has only one solution smaller than three given by

$$\begin{aligned} \ell _J=\frac{3-\mu \sqrt{8+\mu ^2}}{1-\mu ^2}, \quad \mu =\frac{1}{2J}. \end{aligned}$$

(A.7)

Indeed, there holds

$$\begin{aligned} \frac{3}{2}< 3-\frac{\sqrt{8+1/4}}{2}< \frac{3-\mu \sqrt{8+\mu ^2}}{1-\mu ^2} <\frac{3-\mu \sqrt{8\mu ^2+\mu ^2}}{1-\mu ^2}=3. \end{aligned}$$

(A.8)

At the limit where J tends to infinity, we have

$$\begin{aligned} \ell _J=3-\mu \sqrt{8}+3\mu ^2+O\!\left( \mu ^3\right) , \quad \mu =\frac{1}{2J} \end{aligned}$$

(A.9)

and therefore

$$\begin{aligned} R_J\!\left( \ell _J\right) = \frac{\mu ^{-1}}{\sqrt{8}\big (1+O\!\left( \mu \right) \big )} \!\left( \rho -\rho \mu +O\!\left( \mu ^2\right) \right) ^{-1/\mu } \end{aligned}$$

(A.10)

where $\rho =3+\sqrt{8}$. Thus we recover

$$\begin{aligned} \begin{aligned} R_J\!\left( \ell _J\right)&\sim \frac{J}{\sqrt{2}}\rho ^{-2J} \!\left( 1-\mu \right) ^{-1/\mu } \sim \frac{\mathrm {e}}{\sqrt{2}}\, J\rho ^{-2J} \end{aligned}\end{aligned}$$

(A.11)

which completes the proof. $\square$